Uniqueness and trapped modes in the linear problem of the steady flow over a submerged hollow

We consider the linear problem of the steady flow past a submerged hollow of rectangular shape. The fluid is assumed to be inviscid and incompressible and the problem is posed in term of a perturbed stream function vanishing at infinity. By introducing a suitable variational form of the problem, we discuss unique solvability in the case of a supercritical stream at infinity and find sufficient conditions on the hollow's size for the existence of non trivial solutions of the homogenous problem (trapped modes).